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On a conjecture about degree deviation measure of graphs | ||
| Transactions on Combinatorics | ||
| دوره 10، شماره 1، خرداد 2021، صفحه 1-8 اصل مقاله (230.23 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2020.121737.1709 | ||
| نویسندگان | ||
| Ali Ghalavand؛ Ali Reza Ashrafi* | ||
| Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, I. R. Iran | ||
| چکیده | ||
| Let $G$ be an $n-$vertex graph with $m$ vertices. The degree deviation measure of $G$ is defined as $s(G)$ $=$ $\sum_{v\in V(G)}|deg_G(v)- \frac{2m}{n}|,$ where $n$ and $m$ are the number of vertices and edges of $G$, respectively. The aim of this paper is to prove the Conjecture 4.2 of [J. A. de Oliveira, C. S. Oliveira, C. Justel and N. M. Maia de Abreu, Measures of irregularity of graphs, Pesq. Oper., 33 (2013) 383--398]. The degree deviation measure of chemical graphs under some conditions on the cyclomatic number is also computed. | ||
| کلیدواژهها | ||
| irregularity؛ degree deviation measure؛ chemical graph | ||
| مراجع | ||
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[1] M. O. Albertson, The irregularity of a graph, Ars Combin., 46 (1997) 219–225.
[2] A. Ali, E. Milovanović, M. Matejić and I. Milovanović, On the upper bounds for the degree deviation of graphs, J. Appl. Math. Comput., http://dx.doi.org/10.1007/s12190-019-01279-6. [3] H. Abdo, S. Brandt and D. Dimitrov, The total irregularity of a graph, Discrete Math. Theor. Comput. Sci., 16 (2014) 201–206. [4] A. Ghalavand, A. R. Ashrafi and I. Gutman, Extremal graphs for the second multiplicative Zagreb index, Bull. Int. Math. Virtual Inst., 8 (2018) 369–383. [5] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total π−electron energy of alternant hydrocar- bons, Chem. Phys. Lett., 17 ( 1972) 535–538. [6] J. A. de Oliveira, C. S. Oliveira, C. Justel and N. M. Maia de Abreu, Measures of irregularity of graphs, Pesq. Oper., 33 (2013) 383–398. [7] V. Nikiforov, Eigenvalues and degree deviation in graphs, Linear Algebra Appl., 414 (2006) 347–360.
[8] T. Réti and Á. Drégelyi-Kiss, On the generalization of harmonic graphs, Discerete Math. Lett., 1 (2019) 16–20. | ||
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